The main focus of this paper is to investigate bifurcation of critical periods and the existence of isochronous center for a certain Liénard system <Equation ID="Equ56"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1355_Article_Equ56.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \ddot{x}+f(x){\dot{x}}+g(x)=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>x</mi> <mo>¨</mo> </mover> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with f and g of degree <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1355_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\le 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>. By computing the period constants we give weak centers of exact order and parameter condition for the isochronous center. Besides, we prove that at most 4 critical periods can occur from the weak center.

Critical period bifurcation and isochronous center condition for a sextic Liénard system

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发表日期：

2025年8月11日

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Wenyi Wang,
Yangtao Li,
Yusen Wu

摘要

The main focus of this paper is to investigate bifurcation of critical periods and the existence of isochronous center for a certain Liénard system $\begin{aligned} \ddot{x}+f(x){\dot{x}}+g(x)=0, \end{aligned}$ $\begin{matrix} \ddot{x} + f (x) \dot{x} + g (x) = 0, \end{matrix}$ with f and g of degree $\le 6$ $\leq 6$ . By computing the period constants we give weak centers of exact order and parameter condition for the isochronous center. Besides, we prove that at most 4 critical periods can occur from the weak center.